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In this second part of a two-part study, we extend to nonlinear synaptic responses a new framework, called Response Surfaces (RSs), for analyzing, designing, and visualizing spiking neurons and networks. An RS is the transfer function between input and output firing times of a spiking neuron and shows all the patterns of input spike times that fire a neuron at a given time. Here in Part II, we present four mathematically tractable functions to model more realistic post-synaptic-potential waveforms, and we build on the linear RS framework of Part I to create a nonlinear RS framework. We then discuss the qualitative differences between the linear and nonlinear RSs frameworks and revisit problems of Part I using the nonlinear RSs framework: graphing the transfer function of a nonlinear spiking neuron, designing an efficient spiking-XOR gate, analyzing phase-tracking in recurrent SNNs, and developing transfer functions for calculating the output firing times of a spiking neuron given a set of input firing times. For the nonlinear RS framework, we show that the output firing time of a nonlinear spiking neuron is equivalent to the center-of-mass of input spike times (acting as positions) and weights (acting as masses) plus a fixed delay plus a second-order correction. The second-order centroid correction is one of the main differences between the linear and nonlinear RS frameworks. For recurrent networks, the nonlinear RS framework also shows a more stable behavior compared to linear SNNs, owing to the smooth shapes of nonlinear post-synaptic-potential waveforms.

A Framework for Analyzing, Designing, and Visualizing Spiking Neural Networks—Part II: Nonlinear Response Surfaces